A generalized version of a low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate using the concept of noninteger derivatives

Abstract and Applied Analysis

Volumn 2013, Issue , 2013, Pages

  Atangana, Abdon ^a   Ahmed, O Aden ^b   Bildik, Necdet ^c  

^a UNIVERSITY OF THE FREE STATE   (South Africa)

^b TEXAS A AND M UNIVERSITY KINGSVILLE   (United States)

^c CELAL BAYAR UNIVERSITY   (Turkey)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84877282100     PISSN: 10853375     EISSN: 16870409     Source Type: Journal    
DOI: 10.1155/2013/671321     Document Type: Article

References (40)

1. Oldham K. B., Spanier J., The Fractional Calculus 1974 New York, NY, USA Academic Press MR0361633 ZBL0292.26011
2. Daftardar-Gejji V., Jafari H., Adomian decomposition: a tool for solving a system of fractional differential equations. Journal of Mathematical Analysis and Applications 2005 301 2 508 518 10.1016/j.jmaa.2004.07.039 MR2105689 ZBL1061.34003 (Pubitemid 39630973)
3. Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations 2006 Amsterdam, The Netherlands Elsevier MR2218073
4. Podlubny I., Fractional Differential Equations 1999 San Diego, Calif, USA Academic Press MR1658022
5. Caputo M., Linear models of dissipation whose Q is almost frequency independent, part II. Geophysical Journal International 1967 13 5 529 539 10.1111/j.1365-246X.1967.tb02303.x
6. Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 New York, NY, USA John Wiley & Sons MR1219954
7. Samko S. G., Kilbas A. A., Marichev O. I., Fractional Integrals and Derivatives: Theory and Applications 1993 Yverdon, Switzerland Gordon and Breach Science MR1347689
8. Zaslavsky G. M., Hamiltonian Chaos and Fractional Dynamics 2008 Oxford, UK Oxford University Press MR2451371
9. Yildirim A., An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation 2009 10 4 445 450
10. Samko S. G., Kilbas A. A., Maritchev O. I., Integrals and derivatives of the fractional order and some of their applications Nauka i TekhnIka, Minsk 1987
11. Podlubny I., Geometric and physical interpretation of fractional integration and fractional differentiation. Fractional Calculus & Applied Analysis 2002 5 4 367 386 MR1967839 ZBL1042.26003
12. Atangana A., New class of boundary value problems. Information Sciences Letters 2012 1 2 67 76
13. Atangana A., Numerical solution of space-time fractional derivative of groundwater flow equation. 2 Proceedings of the International Conference of Algebra and Applied Analysis June 2012 Istanbul, Turkey 1 20
14. Jumarie G., On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion. Applied Mathematics Letters 2005 18 7 817 826 10.1016/j.aml.2004.09.012 MR2144731 ZBL1075.60068 (Pubitemid 40612599)
15. Jumarie G., Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Computers & Mathematics with Applications 2006 51 9-10 1367 1376 10.1016/j.camwa.2006.02.001 MR2237634 ZBL1137.65001
16. Davison M., Essex C., Fractional differential equations and initial value problems. The Mathematical Scientist 1998 23 2 108 116 MR1738325 ZBL0919.34005
17. Coimbra C. F. M., Mechanics with variable-order differential operators. Annalen der Physik 2003 12 11-12 692 703 10.1002/andp.200310032 MR2020716 ZBL1103.26301
18. Andrianov I., Awrejcewicz J., Construction of periodic solutions to partial differential equations with non-linear boundary conditions. International Journal of Nonlinear Sciences and Numerical Simulation 2000 1 4 327 332 10.1515/IJNSNS.2000.1.4.327 MR1793840 (Pubitemid 33220683)
19. Bender C. M., Milton K. A., Pinsky S. S., Simmons L. M. Jr., A new perturbative approach to nonlinear problems. Journal of Mathematical Physics 1989 30 7 1447 1455 10.1063/1.528326 MR1002247 ZBL0684.34008
20. Delamotte B., Nonperturbative (but approximate) method for solving differential equations and finding limit cycles. Physical Review Letters 1993 70 22 3361 3364 10.1103/PhysRevLett.70.3361 MR1218585 ZBL1051.65505
21. Shariyat M., Automotive Body: Analysis and Design 2006 Tehran, Iran K. N. Toosi University Press
22. Ollson R., Impact response of orthotropic composite plates predicted form a one-parameter differential equation. American Institute of Aeronautics and Astronautics Journal 1992 30 6 1587 1596 10.2514/3.11105
23. Yigit A. S., Christoforou A. P., On the impact of a spherical indenter and an elastic-plastic transversely isotropic half-space. Composites 1994 4 11 1143 1152 10.1016/0961-9526(95)91288-R
24. Yigit A. S., Christoforou A. P., On the impact between a rigid sphere and a thin composite laminate supported by a rigid substrate. Composite Structures 1995 30 2 169 177 10.1016/0263-8223(94)00037-9
25. Christoforou A. P., Yigit A. S., Characterization of impact in composite plates. Composite Structures 1998 43 5 24
26. Christoforou A. P., Yigit A. S., Effect of flexibility on low velocity impact response. Journal of Sound and Vibration 1998 217 3 563 578 10.1006/jsvi.1998.1807 (Pubitemid 128386991)
27. Yigit A. S., Christoforou A. P., Limits of asymptotic solutions in low-velocity impact of composite plates. Composite Structures 2007 81 568 574 10.1016/j.compstruct.2006.10.006 (Pubitemid 46958974)
28. Atangana A., Secer A., A note on fractional order derivatives and Table of fractional derivative of some specials functions. Abstract Applied Analysis. In press
29. Solomon T. H., Weeks E. R., Swinney H. L., Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Physical Review Letters 1993 71 3975 3978 10.1103/PhysRevLett.71.3975
30. Magin R. L., Fractional Calculus in Bioengineering 2006 Connecticut, UK Begell House Publisher
31. Magin R. L., Abdullah O., Baleanu D., Zhou X. J., Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. Journal of Magnetic Resonance 2008 190 255 270 10.1016/j.jmr.2007.11. 007
32. Chechkin A. V., Gorenflo R., Sokolov I. M., Fractional diffusion in inhomogeneous media. Journal of Physics 2005 38 42 L679 L684 10.1088/0305-4470/38/42/L03 MR2186196 ZBL1082.76097 (Pubitemid 41428584)
33. Santamaria F., Wils S., de Schutter E., Augustine G. J., Anomalous diffusion in purkinje cell dendrites caused by spines. Neuron 2006 52 4 635 648 2-s2.0-33751019968 10.1016/j.neuron.2006.10.025 (Pubitemid 44750568)
34. Sun H. G., Chen W., Chen Y. Q., Variable order fractional differential operators in anomalous diffusion modelling. Journal of Physics A 2009 388 4586 4592
35. Atangana A., Secer A., The time-fractional coupled-Korteweg-de-vries equations. Abstract Applied Analysis 2013 2013 8 947986 10.1155/2013/947986
36. Atangana A., Botha J. F., Analytical solution of groundwater flow equation via Homotopy Decomposition Method. Journal of Earth Science & Climatic Change 2012 3 115
37. Shariyat M., Ghajar R., Alipour M. M., An analytical solution for a low velocity impact between a rigid sphere and a transversely isotropic strain-hardening plate supported by a rigid substrate. Journal of Engineering Mathematics 2012 75 107 125 10.1007/s10665-011-9505-1 MR2949534 ZBL1254.74091
38. Awrejcewicz J., Krysko V. A., Saltykova O. A., Chebotyrevskiy Yu. B., Nonlinear vibrations of the Euler-Bernoulli beam subjected to transversal load and impact actions. Nonlinear Studies 2011 18 3 329 364 MR2816175 ZBL1229.37103
39. Poe C. C. Jr., Illg W., Chamis C. C., Strength of a thick graphite/epoxy rocket motor case after impact by a blunt object. Test Methods for Design Allowable for Fibrous Composites 1989 2 Philadelphia, Pa, USA ASTM 150 179. ASTM STP 1003
40. Poe C. C. Jr., Simulated impact damage in a thick graphite/epoxy laminate using spherical indenters. NASA TM 1988 100539